NIERT: Accurate Numerical Interpolation through Unifying Scattered Data Representations using Transformer Encoder

Numerical interpolation for scattered data, i.e., estimating values for target points based on those of some observed points, is widely used in computational science and engineering. The existing approaches either require explicitly pre-defined basis functions, which makes them inflexible and limits their performance in practical scenarios, or train neural networks as interpolators, which still have limited interpolation accuracy as they treat observed and target points separately and cannot effectively exploit the correlations among data points. Here, we present a learning-based approach to numerical interpolation for scattered data using encoder representation of Transformers (called NIERT). Unlike the recent learning-based approaches, NIERT treats observed and target points in a unified fashion through embedding them into the same representation space, thus gaining the advantage of effectively exploiting the correlations among them. The specially-designed partial self-attention mechanism used by NIERT makes it escape from the unexpected interference of target points on observed points. We further show that the partial self-attention is essentially a learnable interpolation module combining multiple neural basis functions, which provides interpretability of NIERT. Through pre-training on large-scale synthetic datasets, NIERT achieves considerable improvement in interpolation accuracy for practical tasks. On both synthetic and real-world datasets, NIERT outperforms the existing approaches, e.g., on the TFRD-ADlet dataset for temperature field reconstruction, NIERT achieves an MAE of $1.897\times 10^{-3}$, substantially better than the state-of-the-art approach (MAE: $27.074\times 10^{-3}$). The source code of NIERT is available at https://anonymous.4open.science/r/NIERT-2BCF.